We first note that (REF ) may be obtained as in the proof of Lemma 5.32 in [1]}. Indeed, the argument in [1]} relies only on Hölder continuity of solutions, Harnack principle, and comparison
principle. The coefficients (in \(\operatorname{BMO}\) ) do not affect the argument since the equation is not used explicitly.
It therefore suffices to show that \(u\)  converges non-tangentially \(\mu \) -a.e. to \(f\) .
For any \(\epsilon >0\) , choose \(f_{\epsilon }\in C_0(\mathbb {R}^n)\)  such that \(\left\Vert f-f_\epsilon \right\Vert _{L^p(\mathbb {R}^n,\mu )}<\epsilon \) . Define \(u_\epsilon (X)=\int _{\mathbb {R}^n}f_\epsilon (y)k(X,y)d\mu (y)\) . Then \(u_\epsilon \in C(\overline{\mathbb {R}_+^{n+1}})\)  and \(u_\epsilon =f_\epsilon \)  on \(\mathbb {R}^n\) .
We note that the latter fact may be gleaned from the analogous fact on bounded domains (see[3]}), the construction at the beginning of Section 2 (applied with \(u=u_\epsilon \) ), and an equicontinuity argument using [3]} Lemma 3.9, and Lemma 4.5.
So
\(\lim _{\Gamma (x)\ni (y,t)\rightarrow (x,0)}u_\epsilon (y,t)=f_\epsilon (x)\qquad \forall \,x\in \mathbb {R}^n.\) 
Since we have the non-tangential convergence for a dense class, the non-tangential convergence of \(u\)  follows from (REF ) and a standard argument. In fact, we have
\(\limsup _{\Gamma (x) \ni (y, t) \rightarrow (x, 0)}|u(y, t)-f(x)| \le \left|N\left(u-u_{\epsilon }\right)(x)\right|+\left|\left(f-f_{\epsilon }\right)(x)\right| \qquad \forall \,x\in \mathbb {R}^n.\) 
For any \(\lambda >0\) , we apply Chebyshev's inequality and (REF ) to get
\(&\mu \left(\left\lbrace x \in \mathbb {R}^{n} : \limsup _{\Gamma (x) \ni (y, t) \rightarrow (x, 0)}|u(y, t)-f(x)|>\lambda \right\rbrace \right)\\&\le \mu \left(\left\lbrace x \in \mathbb {R}^{n} : N\left(u-u_{\epsilon }\right)(x)>\lambda / 2\right\rbrace \right)+\mu \left(\left\lbrace x \in \mathbb {R}^{n} :\left|\left(f-f_{\epsilon }\right)(x)\right|>\lambda / 2\right\rbrace \right)\\&\lesssim \lambda ^{-p}\left(\left\Vert N(u-u_\epsilon )\right\Vert ^p_{L^p(\mathbb {R}^n,d\mu )}+\left\Vert f-f_\epsilon \right\Vert _{L^p(\mathbb {R}^n,d\mu )}^p\right)\\&\lesssim \lambda ^{-p}\left\Vert f-f_\epsilon \right\Vert _{L^p(\mathbb {R}^n,d\mu )}^p\lesssim \epsilon \lambda ^{-p}.\) 

Since \(\epsilon >0\)  is arbitrary, it shows that \(\lim _{\Gamma (x) \ni (y, t) \rightarrow (x, 0)} u(y, t)=f(x)\)  for \(\mu \) - a.e. \(x\in \mathbb {R}^n\) .
